Harmonic Maass form

In mathematics, a weak Maass form is a smooth function  f  on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps.If the eigenvalue of  f  under the Laplacian is zero, then  f  is called a harmonic weak Maass form, or briefly a harmonic Maass form. In mathematics, a weak Maass form is a smooth function  f  on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps.If the eigenvalue of  f  under the Laplacian is zero, then  f  is called a harmonic weak Maass form, or briefly a harmonic Maass form. A weak Maass form which has actually moderate growth at the cusps is a classical Maass wave form. The Fourier expansions of harmonic Maass forms often encode interesting combinatorial, arithmetic, or geometric generating functions. Regularized theta lifts of harmonic Maass forms can be used to construct Arakelov Green functions for special divisors on orthogonal Shimura varieties. A complex-valued smooth function  f  on the upper half-plane  H = {z ∈ C:  Im(z) > 0}  is called a weak Maass form of integral weight k (for the group SL(2, Z)) if it satisfies the following three conditions: If  f  is a weak Maass form with eigenvalue 0 under Δ k {displaystyle Delta _{k}} , that is, if Δ k f = 0 {displaystyle Delta _{k}f=0} , then  f  is called a harmonic weak Maass form, or briefly a harmonic Maass form. Every harmonic Maass form  f  of weight  k  has a Fourier expansion of the form where  q=e2πiz, and  n+,  n−  are integers depending on  f . Moreover, Γ ( s , y ) = ∫ y ∞ t s − 1 e − t d t {displaystyle Gamma (s,y)=int _{y}^{infty }t^{s-1}e^{-t}dt} denotes the incomplete gamma function (which has to be interpreted appropriately when  n=0 ).The first summand is called the holomorphic part, and the second summand is called the non-holomorphic part of  f . There is a complex anti-linear differential operator ξ k {displaystyle xi _{k}} defined by Since Δ k = − ξ 2 − k ξ k {displaystyle Delta _{k}=-xi _{2-k}xi _{k}} , the image of a harmonic Maass form is weakly holomorphic.Hence, ξ k {displaystyle xi _{k}} defines a map from the vector space  Hk  of harmonic Maass forms of weight  k  to the space  M!2-k  of weakly holomorphic modular forms of weight  2-k .It was proved in (Bruinier & Funke 2004) (for arbitrary weights, multiplier systems, and congruence subgroups) that this map is surjective. Consequently, there is an exact sequence